7.2.4.1.1 Example 1 \(xy^{\prime \prime \prime }+\left ( x^{2}-3\right ) y^{\prime \prime }+4xy^{\prime }+2y=0\)

Comparing to standard form \(p_{3}y^{\prime \prime \prime }+p_{2}y^{\prime \prime }+p_{1}y^{\prime }+p_{0}y=f\left ( x\right ) \) shows that

\begin{align*} p_{3} & =x\\ p_{2} & =x^{2}-3\\ p_{1} & =4x\\ p_{0} & =2\\ f\left ( x\right ) & =0 \end{align*}

Checking if it is exact

\begin{align*} p_{3}^{\prime \prime \prime }-p_{2}^{\prime \prime }+p_{1}^{\prime }-p_{0} & =0-2+4-2\\ & =0 \end{align*}

The first integral is therefore

\begin{align*} \frac {d}{dx}\left ( p_{3}y^{\prime \prime }+\left ( p_{2}-p_{3}^{\prime }\right ) y^{\prime }+\left ( p_{1}-p_{2}^{\prime }+p_{3}^{\prime \prime }\right ) y\right ) & =f\left ( x\right ) \\ \frac {d}{dx}\left ( xy^{\prime \prime }+\left ( x^{2}-3-1\right ) y^{\prime }+\left ( 4x-2x+0\right ) y\right ) & =0\\ \frac {d}{dx}\left ( xy^{\prime \prime }+\left ( x^{2}-4\right ) y^{\prime }+2xy\right ) & =0 \end{align*}

Hence the first integral is

\[ xy^{\prime \prime }+\left ( x^{2}-4\right ) y^{\prime }+2xy=c_{1}\]

Let us now check if this is also exact. This has form

\[ p_{2}y^{\prime \prime }+p_{1}y^{\prime }+p_{0}=f\left ( x\right ) \]

Where now

\begin{align*} p_{2} & =x\\ p_{1} & =\left ( x^{2}-4\right ) \\ p_{0} & =2x\\ f\left ( x\right ) & =c_{1}\end{align*}

Checking if it is exact

\begin{align*} p_{2}^{\prime \prime }-p_{1}^{\prime }+p_{0} & =0-2x+2x\\ & =0 \end{align*}

Show it is exact. Therefore its first integral is

\begin{align*} \left ( p_{2}y^{\prime }+\left ( p_{1}-p_{2}^{\prime }\right ) y\right ) ^{\prime } & =f\left ( x\right ) \\ \left ( xy^{\prime }+\left ( \left ( x^{2}-4\right ) -1\right ) y\right ) ^{\prime } & =c_{1}\\ \left ( xy^{\prime }+\left ( x^{2}-5\right ) y\right ) ^{\prime } & =c_{1}\end{align*}

Hence first integral is

\begin{align*} xy^{\prime }+\left ( x^{2}-5\right ) y & =\int c_{1}dx+c_{2}\\ & =c_{1}x+c_{2}\end{align*}

This is first oder linear ode which is now easily solved.